We do something like what I think you are describing in the ASU's aikiken from yokomenuchi (aikiden #1, 2nd and 4th variation) and tsuki (aikiken #6). (Unless they've been re-ordered AGAIN!) We do something like the sliding tap down in aikiken number #5 aswell, but we don't go for a shomen to the head next. Aikiken #1 variation 5 and aikiken #10 also have a sliding down uke's sword into a shomen, but the attack you are defending is from a low position coming up toward your neck, and your sword is actually under theirs to start that movement. I can't think of anythng exactly matching that description in kashima either. What style of sword were you doing?

When you do this, do you tap their sword down with the flat (non-sharp) "side" of your sword? And do you primarily make the power happen by using your back hand?